It is known that the field of 3D printing technology by photo-curing can comprise two basic technologies: stereolithographic printing, in which a laser emitting around 400 nm is used, to solidify by means of the beam emitted a photo-curing polymer in the liquid state which is in a special tank; DLP printing (Digital Light Processing), according to which a photo-curing polymer, again in the liquid state in a tank, is exposed to the luminous radiation emitted by a device similar to a projector.
According to both these technologies, the printing process proceeds making one layer after another, that is, solidifying a first layer adhering to a supporting plate (or extraction plate) and then a second layer adhering to the first layer and so on until formation of the complete object. Therefore, according to this technology, the data representing the three-dimensional object to be formed are organised as a series two-dimensional layers which represent transversal sections of the object.
According to the Bottom-Up method, applied to machines both of the SLA and DLP type, the plate for extracting the object moves from the bottom upwards, with a layer-by-layer tilting movement.
Basically, the method for formation of three-dimensional objects comprises:                a software subdivides the 3D model, provided as input for the printing, in an ordered succession of layers, with the thickness determined according to the technology adopted, the opacity of the polymer, the quantity of the catalyst, the degree of precision to be obtained and the characteristics of the machine provided, usually between 50 and 200 microns, but in any case a succession of a discrete and finite number of layers;        an extraction plate, consisting of a material which is able to facilitate the gluing on itself of the first layer of polymer, moves to a predetermined distance from the first layer and waits for the light beam (SLA or DLP) to solidify the first layer; it then raises by a distance sufficient for the layer just formed to detach from the base of the tank (usually approx. 1 mm) and then lowers by the same distance, less the predetermined distance for the formation of the second layer, and so on until forming the entire object.        
The resulting to and fro movement, also called the tilting movement, has two main purposes: it allows the layer just formed to detach from the base of the tank, and at the same time it allows a new quantity of liquid resin not polymerised to interpose between the layer just formed and the base of the container, to allow the refreshing of material still in the liquid state beneath the layer already solidified, for the curing and the formation of the next layer.
In order to summarise the above-mentioned system with a mathematical model it is possible to identify the following parameters:
tc=exposure time of the luminous beam for curing the polymer (a function in particular of the electromagnetic power of the spectrum of interest dispensed by the luminous source and the ambient working conditions, absence of oxygen, intensity and covering capacity of the colour of the polymer, quantity of catalyst)
s=thickness of layer
n=number of layers formed
d=tilting distance of the extraction plate
h=height of object to be printed
tb=tilting time
vb=tilting speed
T=total time for printing three-dimensional object from which it may be deduced that the number n of layers to be hardened will be equal to the ratio between the height h of the object to be printed and the thickness s of the layer to be solidified, according to the equation:n=h/s 
The tilting time is defined as the time taken for the extraction plate to travel the distance d+(d−s), that is, the distance of rising of the plate for the renewal or refreshing of the resin, added to the distance for return to the printing position, decreased by the thickness of the layer to be produced, which gives:tb=(d+(d−s))/vb which means that the printing process requires a total time of:
  T  =            n      *              (                  t          +                      t            b                          )              =                  h        s            *              (                  t          +                                    d              +                              (                                  d                  -                  s                                )                                                    v              b                                      )            
This equation represents, in fact, the characteristic equation of the printing time of the DLP technology, which may be enriched with further elements, such as, for example, a differentiation between the tilting speed when raising and when lowering (it tends to slow down during the detaching phase when raising to avoid too high a mechanical stress of the layer just formed), the rest time, to be added subsequently to the light exposure time of the layer to allow the resin, as soon as it is solidified, to consolidate the interactions and therefore be more resistant to the mechanical stress which tends to detach it from the base, and so on.
With regard to the characteristic equation of the printing time for the stereolithographic printing technology, it should be noted that there is a substantial difference with respect to the equation relative to the DLP technology. It is evident, in fact, how the DLP technology depends exclusively on the height of the object to be printed and not on its shape or its volume, given that the surface of the layer which is formed is all projected simultaneously, which is not the case for the stereolithographic technology. In effect, with the same power dispensed by the light sources in the two cases, the stereolithographic technology uses a laser, which takes time to scan the entire surface of the single layer to be produced.
The characteristic equation of the printing time for the stereolithographic printing technology will therefore also be dependent on the volume of the object to be printed.
Consequently, where
Sn=surface of Nth layer;
Ss=scanning surface of laser beam;
tp=time of persistence of the laser beam on the unit of surface Ss;
tl=latency time, that is, the time taken by the laser beam to move on the next Ss;
the curing time of the single layer is:(Sn/Ss)*(tp+tl)and therefore, by summing all the n layers:
      ∑    1    n    ⁢          ⁢                    S        n                    S        s              *          (                        t          p                +                  t          l                    )      from which the characteristic equation of the printing time of the stereo lithographic printing technology is:
  T  =                    (                  n          *                      t            b                          )            +                        ∑          1          n                ⁢                                  ⁢                                            S              n                                      S              s                                *                      (                                          t                p                            +                              t                l                                      )                                =                            h          s                *                              d            +                          (                              d                -                s                            )                                            v            b                              +                        ∑          1          n                ⁢                                  ⁢                                            S              n                                      S              s                                *                      (                                          t                p                            +                              t                l                                      )                              which, as already mentioned, is dependent not only on the height of the object to be formed but also on its volume.
Consequently, whilst the printing time for the DLP printing technology, indicated with TDLP, in order to simultaneously produce two objects of equal height would not change in any way, the printing time for the stereolithographic printing technology, indicated with TSLA, would be dependent (less the tilting time, to be considered equal) on the sum of the two single characteristic printing times of the two objects.
The previous paragraphs have dealt with two of the three main issues concerning the three-dimensional printing process by photo-curing, the one connected to the mechanical management aspects, which are of interest to the formation in successive layers, and the one relative to the time equations which determine the characteristics of the printing dynamics management software.
The third issue, which is no less important, concerns the characteristics of the resin collection system, the so-called tank, which has the purpose of not merely containing the liquid polymer from which the printed three-dimensional object is obtained by photo-curing, but also facilitating the formation and the detachment of the layer which has just been formed, and facilitating the gluing towards the extraction plate, without the mechanical strength adversely affecting the integrity.
By analysing the characteristics of the prior art solutions it is possible to summarise the bottom-up collection systems, both for the DLP and SLA technologies, as follows:                container of the resin, with hollow base;        material transparent to UV rays for covering the base;        layer of non-stick material for covering the transparent material.        
A hole is made, usually at the centre of the collection system, to allow the passage of the light beam for triggering the photo-curing phenomenon; the hole is covered with glass which has excellent UV ray transparency characteristics (in order not to lose incident luminous power), such as, for example, quartz and borosilicate glass. Lastly, the most important part to allow the correct performance of the process certainly concerns the coverage of the glass with a non-stick material, to allow the first layer to adhere to the extraction plate and the successive layers to join together in sequence.
The failure of this process would result in the falling of the layer just formed onto the base of the tank, interrupting the forming process and causing the failure of the printing routine.
The limiting effects of this technology, which render the production of the object very slow (up to hours per centimeter), very unstable and with the capacity to make objects with small dimensions, are investigated below.